Two rods of different materials having coefficients of thermal expansion `alpha_(1) , alpha_(2)` and Young's Moduli `Y_(1), Y_(2)` respectively are fixed between two rigid massive walls. The rods are heated such that they undergo the same increase in temperature. There is no bending of the rods. If `alpha_(1) : alpha_(2) = 2 : 3,` the thermal stresses developed in the two rods are equal provided `Y_(1) : Y_(2)` is equal to

Options

A. `2:3`
B. `1:1`
C. `3:2`
D. `4:9`

Explanation

Expansion in the rod due to rise in temp Compression in rod. For the first rod `Deltal_(1)=alpha_(1)l_(1)Deltatheta`
Compression in the rod `Deltal_(1)= (-F)/(A) (l_(1))/(Y_(1))`
or the length of the rod remains unchanged
`alpha_(1)l_(1)Delta theta= (-Fl_(1))/(AY_(1))`
or
`alpha_(1) Delta theta = (-F)/(AY_(1))--- (i)`
Similarly for second rod
`alpha_(2) Delta theta = (-F)/(AY_(2))--- (ii)`
`(alpha_(1))/(alpha_(2))=(Y_(2))/(Y_(1))or(Y_(1))/(Y_(2))=(alpha_(2))/(alpha_(1))=(3)/(2)`